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A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

\[\cos\left( \frac{\pi}{11}\right) - \cos\left( \frac{2\pi}{11} \right)+ \cos\left( \frac{3\pi}{11}\right) \\ - \cos\left( \frac{4\pi}{11} \right)+ \cos\left( \frac{5\pi}{11} \right)=\ ?\]

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Given \(z^2+z+1=0,\) find the value of

\[\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.\]

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\[ \large{\dfrac{31-2\delta_{1}}{1-2\delta_{1}} +\dfrac{31-2\delta_{2}}{1-2\delta_{2}}+\dfrac{31-2\delta_{3}}{1-2\delta_{3}}} \]

If \(\large{1,\delta_{1},\delta_{2},\delta_{3}}\) are distinct fourth roots of unity, then evaluate the expression above.

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If \(w=e^{2\pi{i}/5}\), find \[\sum_{k=1}^{4}\frac{1}{1+w^k+w^{2k}}\]

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